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Physical Review A

First-degree homogeneousN-particle noninteracting kinetic-energy density functionals

Tamás Gál

Published Nov 16, 2001• 14 citations
Publisher

Abstract

It is known in density-functional theory that the noninteracting kinetic-energy density functional ${T}_{s}[\ensuremath{\rho}]$ is not first-degree homogeneous in density scaling. However, it is shown here that, for every particle number N, there is an N-particle noninteracting kinetic-energy density functional ${T}_{N}[\ensuremath{\rho}],$ that is, a density functional that gives the noninteracting kinetic energy for N-particle densities, which is of first-degree homogeneity in the density $\ensuremath{\rho}(\stackrel{\ensuremath{\rightharpoonup}}{r}).$ This gives a powerful tool, a strong requirement, for constructing such functionals. A systematic procedure to obtain the real part of ${T}_{N}[\ensuremath{\rho}],$ the full ${T}_{N}[\ensuremath{\rho}]$ in one-dimension, for each N is also proposed. It is pointed out, further, that in the Euler-Lagrange equations that determine the one-particle orbitals that define ${T}_{s}[\ensuremath{\rho}],$ the Lagrange multiplier that forces the orbitals to yield $\ensuremath{\rho}(\stackrel{\ensuremath{\rightharpoonup}}{r})$ is not other than the first derivative of ${T}_{s}[\ensuremath{\rho}],$ $\ensuremath{\delta}{T}_{s}[\ensuremath{\rho}]/\ensuremath{\delta}\ensuremath{\rho}(\stackrel{\ensuremath{\rightharpoonup}}{r}),$ which yields a natural derivation of the Kohn-Sham equations. Utilizing the same idea, it is shown for ground states how the Schr\"odinger equation can be derived from the basics of density-functional theory as well.

DOI
10.1103/physreva.64.062503

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It is known in density-functional theory that the noninteracting kinetic-energy density functional Ts[\ensuremathρ]{T}_{s}[\ensuremath{\rho}]Ts​[\ensuremathρ] is not first-degree homogeneous in density scaling. However, it is shown here that, for every particle number N, there is an N-particle noninteracting kinetic-energy density functional TN[\ensuremathρ],{T}_{N}[\ensuremath{\rho}],TN​[\ensuremathρ], that is, a density functional that gives the noninteracting kinetic energy for N-particle densities, which is of first-degree homogeneity in the density \ensuremathρ(r\ensuremath⇀).\ensuremath{\rho}(\stackrel{\ensuremath{\rightharpoonup}}{r}).\ensuremathρ(r\ensuremath⇀). This gives a powerful tool, a strong requirement, for constructing such functionals. A systematic procedure to obtain the real part of TN[\ensuremathρ],{T}_{N}[\ensuremath{\rho}],TN​[\ensuremathρ], the full TN[\ensuremathρ]{T}_{N}[\ensuremath{\rho}]TN​[\ensuremathρ] in one-dimension, for each N is also proposed. It is pointed out, further, that in the Euler-Lagrange equations that determine the one-particle orbitals that define Ts[\ensuremathρ],{T}_{s}[\ensuremath{\rho}],Ts​[\ensuremathρ], the Lagrange multiplier that forces the orbitals to yield \ensuremathρ(r\ensuremath⇀)\ensuremath{\rho}(\stackrel{\ensuremath{\rightharpoonup}}{r})\ensuremathρ(r\ensuremath⇀) is not other than the first derivative of Ts[\ensuremathρ],{T}_{s}[\ensuremath{\rho}],Ts​[\ensuremathρ], \ensuremathδTs[\ensuremathρ]/\ensuremathδ\ensuremathρ(r\ensuremath⇀),\ensuremath{\delta}{T}_{s}[\ensuremath{\rho}]/\ensuremath{\delta}\ensuremath{\rho}(\stackrel{\ensuremath{\rightharpoonup}}{r}),\ensuremathδTs​[\ensuremathρ]/\ensuremathδ\ensuremathρ(r\ensuremath⇀), which yields a natural derivation of the Kohn-Sham equations. Utilizing the same idea, it is shown for ground states how the Schr"odinger equation can be derived from the basics of density-functional theory as well.
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The State--Specific Kohn--Sham Density Functional Theory [arXiv:physics/0506037] is used to derive the Kohn-Sham exchange-correlation potential $\vxc$ and exchange-correlation energy $\Eco$ as explicit functionals of vsv_svs​ and ϕ1ϕ_1ϕ1​, where vsv_svs​ is the local, one-body potential from the Kohn--Sham equations, and ϕ1ϕ_1ϕ1​ is the spinless one-particle density matrix from the Kohn--Sham noninteracting state, say $|ϕ_1\ran$. In other words, $|ϕ_1\ran$ is the ground state eigenfunction of the noninteracting Schrödinger equation with the one-body potential vsv_svs​. For simplicity, we only consider noninteracting states that are closed-shell states and interacting states that are nondegenerate, singlet ground-states.
Explicit kinetic-energy density functionals for noninteracting electron systems that are based on reproducing the linear response function in the homogeneous limit are the most accurate approximations to the exact Kohn-Sham functional. In this work we study several such models for one-dimensional systems. In this case, the geometrical simplicity allows the calculation of the static linear response resulting from these functionals for inhomogeneous systems. Likewise, the discussion of the energies and density profiles obtained is easier and clearer than that for more realistic problems. Special attention is paid to the application of explicit kinetic energy functionals to systems with odd and noninteger number of particles.
In earlier work, expressions have been constructed for the single-particle kinetic-energy functional ${T}_{s}[\ensuremath{\rho}]$ for independent fermions subject to harmonic confinement in low dimensions, with $\ensuremath{\rho}$ the particle density. Here, the differential equation for $\ensuremath{\rho}$ is first obtained in d dimensions for an arbitrary number of closed shells. Then, by using the known Euler-Lagrange equation, the functional derivative $\ensuremath{\delta}{T}_{s}/\ensuremath{\delta}\ensuremath{\rho}(\mathbf{r})$ is constructed. ${T}_{s}[\ensuremath{\rho}]$ itself is proved to take the form of a linear combination of three pieces: (i) a von Weizs\"acker inhomogeneity kinetic energy, but with the original coefficient reduced by a dimensionality factor 1/d,1/d,1/d, (ii) a Thomas-Fermi kinetic energy, and (iii) a truly nonlocal contribution that, however, is shown to involve only the density $\ensuremath{\rho}$ itself and its first derivative. Thus, for this model, which is currently highly relevant to the interpretation of experiments on the evaporative cooling of dilute, and hence almost noninteracting, fermions, a complete density-functional theory now exists.
By utilizing the knowledge that a Hamiltonian is a unique functional of its ground-state density, the following fundamental connections between densities and Hamiltonians are revealed: Given that ${\ensuremath{\rho}}_{\ensuremath{\alpha}}, {\ensuremath{\rho}}_{\ensuremath{\beta}},\dots{},{\ensuremath{\rho}}_{\ensuremath{\omega}}$ are ground-level densities for interacting or noninteracting Hamiltonians H1,H2,…,HM{H}_{1}, {H}_{2},\dots{},{H}_{M}H1​,H2​,…,HM​ (MMM arbitrarily large) with local potentials v1{v}_{1}v1​,v2{v}_{2}v2​,…\dots{}…,vM{v}_{M}vM​, but given that we do not know which $\ensuremath{\rho}$ belongs with which HHH, the correct mapping is possible and is obtained by minimizing $\ensuremath{\int}d\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}} [{v}_{1}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\alpha}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})+{v}_{2}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\beta}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})+\ensuremath{\cdots}{v}_{M}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}}){\ensuremath{\rho}}_{\ensuremath{\omega}}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{r}})]$ with respect to optimum permutations of the $\ensuremath{\rho}$'s among the vvv's. A tight rigorous bound connects a density to its interacting ground-state energy via the one-body potential of the interacting system and the Kohn-Sham effective one-body potential of the auxiliary noninteracting system. A modified Kohn-Sham effective potential is defined such that its sum of lowest orbital energies equals the true interacting ground-state energy. Moreover, of all those effective potentials which differ by additive constants and which yield the true interacting ground-state density, this modified effective potential is the most invariant with respect to changes in the one-body potential of the true Hamiltonian. With the exception of the occurrence of certain linear dependencies, aaa density will not generally be associated with any ground-state wave function (is not wave function vvv representable) if that density can be generated by a special linear combination of three or more densities that arise from a common set of degenerate ground-state wave functions. Applicability of the "constrained search" approach to density-functional theory is emphasized for non-vvv-representable as well as for vvv-representable densities. In fact, a particular constrained ensemble search is revealed which provides a general sufficient condition for non-vvv representability by a wave function. The possible appearance of noninteger occupation numbers is discussed in connection with the existence of non-vvv representability for some Kohn-Sham noninteracting systems.
Using an approach based on many body perturbation theory, the correlation energy $\cEco$ is expressed as an explicit functional of ρ1ρ_1ρ1​, vvv, and vsv_svs​, where ρ1ρ_1ρ1​ is the one-particle density matrix from the noninteracting, or reference, determinantal-state; vvv is the external potential from the interacting, or target, state; vsv_svs​ is the (kernel of the) external potential from the noninteracting determinantal-state. In other words we have $\cEco[ρ_1,v,v_s]$. Anther possibility is the following explicit functional: $\cEco[ρ_1,v_{\text{co}},v_s]$, where vcov_{\text{co}}vco​ is the (kernel of the) correlation potential from the noninteracting Hamiltonian. The proposed method can, in principle, be used to compute $\cEco$ in a very accurate and efficient manner, since, like the Kohn--Sham approach, there are no virtual orbitals to consider. However, in contrast to the Kohn--Sham approach, $\cEco$ is a known, explicit functional that can be approximated in a systematic manner. For simplicity, we only consider noninteracting closed-shell states and target states that are nondegenerate, singlet ground-states; so, in that case, ρ1ρ_1ρ1​ denotes the spin-less one-particle density matrix from the determinantal reference state.